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LOWER-ORDER TERMS OF THE ONE-LEVEL DENSITY OF A FAMILY OF QUADRATIC HECKE $\boldsymbol {L}$-FUNCTIONS

Part of: Discontinuous groups and automorphic forms Exponential sums and character sums Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  22 February 2022

PENG GAO  and
LIANGYI ZHAO Open the ORCID record for LIANGYI ZHAO [Opens in a new window]
PENG GAO
Affiliation:
School of Mathematical Sciences, Beihang University, Beijing 100191, China e-mail: [email protected]
LIANGYI ZHAO*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
*
e-mail: [email protected]
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Abstract

In this paper, we study lower-order terms of the one-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the generalized Riemann hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$ . Moreover, we apply the ratios conjecture of L-functions to derive these lower-order terms as well. Up to the first lower-order term, we show that our results are consistent with each other when the Fourier transforms of the test functions are supported in $(-2, 2)$ .

Keywords

one-level densitylow-lying zerosquadratic Hecke L-functions

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 11L40: Estimates on character sums 11M06: $zeta (s)$ and $L(s, chi)$ 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M50: Relations with random matrices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 114 , Issue 2 , April 2023 , pp. 178 - 221
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Dzmitry Badziahin

P. G. is supported in part by NSFC grant 11871082 and L. Z. by FRG grant PS43707 and the Goldstar Award PS53450 from the University of New South Wales (UNSW). Parts of this work were done when P. G. visited UNSW in September 2019. He wishes to thank UNSW for the invitation, financial support and warm hospitality during his pleasant stay. Finally, the authors thank the anonymous referee for his/her very careful reading of this manuscript and many helpful comments and suggestions.

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